# Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems

Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous... We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly $$(p-1)$$ ( p - 1 ) -sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value $$\lambda _*>0$$ λ ∗ > 0 such that for all $$\lambda >\lambda _*$$ λ > λ ∗ the problem has at least two positive solutions, for $$\lambda =\lambda _*$$ λ = λ ∗ it has at least one positive solution and for $$\lambda \in (0,\lambda _*)$$ λ ∈ ( 0 , λ ∗ ) there are no positive solutions. Also, for $$\lambda \ge \lambda _*$$ λ ≥ λ ∗ we show that the problem has a smallest positive solution $$\bar{u}_{\lambda }$$ u ¯ λ and we investigate the continuity and monotonicity properties of the map $$\lambda \rightarrow \bar{u}_{\lambda }$$ λ → u ¯ λ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

# Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems

, Volume 154 (2) – Feb 9, 2017
18 pages

/lp/springer_journal/existence-nonexistence-and-multiplicity-of-positive-solutions-for-EYuBxdUyza
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-017-0919-6
Publisher site
See Article on Publisher Site

### Abstract

We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly $$(p-1)$$ ( p - 1 ) -sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value $$\lambda _*>0$$ λ ∗ > 0 such that for all $$\lambda >\lambda _*$$ λ > λ ∗ the problem has at least two positive solutions, for $$\lambda =\lambda _*$$ λ = λ ∗ it has at least one positive solution and for $$\lambda \in (0,\lambda _*)$$ λ ∈ ( 0 , λ ∗ ) there are no positive solutions. Also, for $$\lambda \ge \lambda _*$$ λ ≥ λ ∗ we show that the problem has a smallest positive solution $$\bar{u}_{\lambda }$$ u ¯ λ and we investigate the continuity and monotonicity properties of the map $$\lambda \rightarrow \bar{u}_{\lambda }$$ λ → u ¯ λ .

### Journal

Manuscripta MathematicaSpringer Journals

Published: Feb 9, 2017

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